Theresa May: she staggers into the summer break with Brexit mired in stalemate.
Photograph: POOL/Reuters

The parliamentary year ends not with a bang, but with a whimper of relief from Downing Street that Mrs May is staggering into the summer break still able to call herself prime minister, a howl of indignation from opposition parties about government cheating in key Commons divisions and an almost-universal groan of despair that Britain is mired in Brexit stalemate. Take back control, they said. How is that working out for you?

The prime minister’s Chequers plan is barely a fortnight old, but already being read the last rites. It is too soft for the hard Brexiters. It is too hard for those who want a soft Brexit. It is being picked apart by a disdainful EU. There is a constituency in parliament for some kind of halfway house arrangement, but Mrs May’s version hasn’t got many enthusiastic takers, even among those MPs willing to entertain compromise.

The febrility and skulduggery of recent days in parliament flow from what has been the fundamental conundrum from the start. That is finding a form of Brexit that satisfies a majority of the Conservative party, a majority of MPs in parliament, a majority in the country and is acceptable to the EU. It is now conceivable that there is no majority in parliament for any kind of deal that Mrs May is capable of striking.

When there is stalemate, folk will look to find ways to resolve it. Saying to themselves: “We can’t carry on like this”, they search for an escape from this hot mess.

One way to attempt to break the deadlock would be to try a new prime minister. A new face at Number 10 might – it would rather depend on the identity of the face – just might begin with more goodwill and credibility. The ultimate deciders of a Tory leadership contest would be the party’s small and largely elderly membership. Given their views, they would be highly likely to replace Mrs May with a Brexiter. With the benefit of hindsight, it might have been better had one of the Outers taken over when David Cameron resigned two years ago. This could have forced the Brexiters to take full possession of the enterprise rather than shooting down everyone else’s suggestions for avoiding a calamitous Brexit while never producing a plausible plan of their own.

The complete absence of such a plan was a conspicuous feature of Boris Johnson’s hollow resignation speech. Something else was also missing. Although the only consistent feature of his career is the throbbing ache of his ambition, there was no call from the former foreign secretary for the removal of Mrs May. The hard Brexiters are ever willing to wound, but afraid to strike. Their noisy bluster has a cowardly heart. The bulk of Conservative MPs continue to calculate that it would be a disaster to create a three-month hiatus in the negotiations with the EU while the Tories conduct a vicious and chaotic fight for the premiership. At the end of it, whoever emerged as leader would face unchanged party and parliamentary arithmetic.

‘The complete absence of such a plan was a conspicuous feature of Boris Johnson’s hollow resignation speech.’ Photograph: HO/AFP/Getty Images

An alternative solution to the stalemate is to assemble a GNU, also known as a government of national unity. This notion has received increased attention because its advocates include Sir Nicholas Soames, the Conservative MP who is the grandson of Sir Winston Churchill. Churchill’s wartime government saw the Tory, Labour and Liberal parties governing together in a joint mission to defeat an external enemy. It is therefore no template for addressing a crisis brought on by Britain’s highly charged internal divisions. The country is not at war with hostile powers, but in conflict with itself. A slightly better precedent for a cross-party cabinet to navigate Brexit might be the “national” government of the 1930s, which was formed after the Wall Street crash to deal with the economic emergency of the Great Depression.

Ramsay MacDonald’s decision to enter a coalition with the Tories split Labour and saw the party eviscerated at the ballot box. MacDonald became an eternal traitor in Labour folklore. Even if Jeremy Corbyn were not mindful of that, he feels neither obligation nor incentive to help the Conservatives extract themselves from the quicksands of Brexit. You don’t throw a rope to your political opponent when he is drowning in a bog of his own making. Brexit is currently “owned” by the Tories, as will be the economic consequences if they prove to be bad. Mr Corbyn is not going to take co-parentage of Brexit. Even centrist Labour MPs who disagree with him about almost everything else will not do the Tories that favour. You are as likely to see a GNU as you are a dodo.

A fresh election would be a third way to try to end the impasse. If the parliament produced by the last election is incapable of resolving Brexit, then perhaps we should ask the people to choose a new parliament. This is the solution desired by the Labour leadership. One problem is that the people might just return another hung parliament, which would leave us back at square one. The opinion polls had been registering modest Tory leads for a few months. In reaction to the most recent trouble and strife, the polls have shifted to report a modest Labour lead. That doesn’t give Conservative MPs any incentive to trigger an early election and an early election can only happen if there are at least some Conservative MPs willing to vote for it.

This might be different were Labour led by a different kind of leader. I guess then there might be some Tories who would be sanguine about the idea of handing over the Brexit torture to the other party. They might even welcome it, but I have yet to meet a Conservative MP who is willing to trigger an election that could put Mr Corbyn in Number 10 and hand control of the Treasury to John McDonnell.

‘I have yet to meet a Conservative MP who is willing to trigger an election that could put Mr Corbyn in Number 10 and hand control of the Treasury to John McDonnell.’ Photograph: Leon Neal/Getty Images

Time is running out on the Brexit negotiations. No one thinks a deal can be done to the supposed October deadline. This has some people talking about buying more time. Another idea being canvassed is to seek more space to find a solution by asking for a postponement of the exit date. The article 50 process can be extended, but only if the other 27 EU member states agree to it. And their agreement has to be unanimous. Everyone would have to give the nod to more time, every country from Cyprus to Slovenia, from Malta to Estonia. One or more EU countries might well treat a request for more time as an opportunity to gain some leverage, a chance to press an agenda or pursue an interest. It is just about conceivable that some injury time could be granted, but it is reckless to assume that this is guaranteed.

The fifth and most dramatic way to break the deadlock is to hand responsibility back to the public by holding a further referendum. The voters created the conditions for this stalemate by narrowly choosing Brexit two long years ago and then declining to give anyone a parliamentary mandate in the subsequent general election. What the people have done, only the people can fix.

The bad objection to a further referendum is that it would be a negation of the people’s will. That’s nonsense. You aren’t negating the people’s will by allowing the people a further opportunity to express their will. If the people choose to change their view, that’s democracy for you. Democracy ceases to be democracy if the people can’t change their minds.

The better objection to another referendum is that it would be hard to agree on the question when there will be three options: to leave on the terms that the government strikes with the EU (if it manages to agree terms); to leave without any deal; or to stay in the EU.

One solution is to put all three to the people, eliminate the option that proved least popular in an initial vote and then ask the people to decide between the two outstanding choices. This could be done by preferential voting. Or it could be done French-style, with two rounds of voting. The idea of the triple-choice plebiscite has been kicked around in nerdy circles for a while and has gained a bit more traction since it was advocated by the former cabinet minister Justine Greening.

This would be a highly contentious constitutional innovation for Britain. It would mean a highly unusual and unpredictable three-sided campaign. For there to be any chance of it happening, at least one of the major parties would have to support the idea and neither is willing to do so at the moment.

That this is being talked about at all tells us just how desperate the Brexit deadlock has become.